Let’s solve the problems presented in Question 9 step by step. Here's the detailed breakdown:

Problem 9

Two vectors are given:
$\vec{a} = (4.0 \, \text{m})\hat{i} - (3.0 \, \text{m})\hat{j} + (1.0 \, \text{m})\hat{k}$
$\vec{b} = (-1.0 \, \text{m})\hat{i} + (1.0 \, \text{m})\hat{j} + (4.0 \, \text{m})\hat{k}.$

You are tasked to find:
(a) $\vec{a} + \vec{b}$,
(b) $\vec{a} - \vec{b}$, and
(c) a vector $\vec{c}$ such that $\vec{a} - \vec{b} + \vec{c} = 0$.

(a) Find $\vec{a} + \vec{b}$:

To add two vectors, sum their corresponding components.
[ \vec{a} + \vec{b} = [(4.0 + (-1.0))\hat{i}] + [(-3.0 + 1.0)\hat{j}] + [(1.0 + 4.0)\hat{k}] ]

Simplify:
$\vec{a} + \vec{b} = (3.0 \, \text{m})\hat{i} - (2.0 \, \text{m})\hat{j} + (5.0 \, \text{m})\hat{k}.$

(b) Find $\vec{a} - \vec{b}$:

To subtract two vectors, subtract their corresponding components.
[ \vec{a} - \vec{b} = [(4.0 - (-1.0))\hat{i}] + [(-3.0 - 1.0)\hat{j}] + [(1.0 - 4.0)\hat{k}] ]

Simplify:
$\vec{a} - \vec{b} = (5.0 \, \text{m})\hat{i} - (4.0 \, \text{m})\hat{j} - (3.0 \, \text{m})\hat{k}.$

(c) Find $\vec{c}$ such that $\vec{a} - \vec{b} + \vec{c} = 0$:

Rearranging for $\vec{c}$:
$\vec{c} = -(\vec{a} - \vec{b}).$

From part (b), we know:
$\vec{a} - \vec{b} = (5.0 \, \text{m})\hat{i} - (4.0 \, \text{m})\hat{j} - (3.0 \, \text{m})\hat{k}.$

Negating each component:
$\vec{c} = -(5.0 \, \hat{i} - 4.0 \, \hat{j} - 3.0 \, \hat{k}) = (-5.0 \, \text{m})\hat{i} + (4.0 \, \text{m})\hat{j} + (3.0 \, \text{m})\hat{k}.$

Final Answers:

1. (a) $\vec{a} + \vec{b} = (3.0 \, \text{m})\hat{i} - (2.0 \, \text{m})\hat{j} + (5.0 \, \text{m})\hat{k}$,
2. (b) $\vec{a} - \vec{b} = (5.0 \, \text{m})\hat{i} - (4.0 \, \text{m})\hat{j} - (3.0 \, \text{m})\hat{k}$,
3. (c) $\vec{c} = (-5.0 \, \text{m})\hat{i} + (4.0 \, \text{m})\hat{j} + (3.0 \, \text{m})\hat{k}$.

Would you like me to solve Question 10 or 11 next? Let me know!

Related Questions:

1. How do you calculate the magnitude of a vector given its components?
2. How do you subtract vectors graphically and analytically?
3. What is the resultant of multiple vectors in unit-vector notation?
4. How can you determine a vector that balances a given vector equation?
5. How do you express vectors in different coordinate systems?

Tip: Always add or subtract vectors component-wise: sum or subtract the coefficients of $\hat{i}$, $\hat{j}$, and $\hat{k}$ separately.